One of the more troubling symptoms suffered by glaucoma patients is a significant increase in intraocular pressure. This increase in pressure may lead to eye damage and even blindness in some patients. One means for relieving the intraocular pressure is to create an alternative outflow channel in the patients eye through a sclerostomy procedure. In that procedure, a fistula is created in the sclera at the peripheral regions of the cornea. The fistula allows liquid aqueous humor produced inside the eye to drain into subconjunctival space and, hence, decrease the volume of liquid contained within the eye. The decrease in volume of liquid results in a corresponding decrease in intraocular pressure. From the subconjunctival space, the liquid is gradually absorbed or translocated away from the interior of the eye. One sclerostomy approach is presented in application Ser. No. 07/356,885 filed by Hsia et al. entitled, "Method and Laser Apparatus for Creating a Fistula in the Sclera of the Eye". That approach involved dyeing a target area so that it would absorb laser energy. Once the area was appropriately dyed, laser energy was directed to reflect off a goniolens through the cornea to the target area.